The uniform primality conjecture for elliptic curves

Abstract

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed

Algebraic Divisibility Sequences Over Function Fields

“…Building on the heuristic argument above, Einsiedler et al [10] conjectured that an EDS has only finitely many prime terms, and this conjecture was later expanded upon by Everest et al [13]. For some EDSs, finiteness follows from a type of generic factorization not unlike (2.1) (see, for example, [13,14,16,26] and Section 6), but the general case appears difficult.…”

“…More generally, most integer Lucas sequences are expected to have infinitely many prime terms [8,17,25]. The only obvious exceptions occur with a type of generic factorization [13]. For example, if f and g are positive coprime integers, then the Lucas sequence associated to f 2 and g 2 ,…”

“…For some EDSs, finiteness follows from a type of generic factorization not unlike (2.1) (see, for example, [13,14,16,26] and Section 6), but the general case appears difficult. The study of primitive divisors in integral Lucas sequences goes back to the 19th-century work of Bang [2] and Zsigmondy [57], who showed that a n − b n has a primitive divisor for all n > 6.…”

“…For example, a longstanding conjecture says that the Mersenne sequence M n = 2 n − 1 contains infinitely many primes, and more generally it is expected that a Lucas sequence has infinitely many prime terms unless it has a 'generic' factorization [8,13,17,25]. On the other hand, because of the rapid growth rate of EDSs, which satisfy log |D n | n 2 , the prime number theorem suggests that EDSs should contain only finitely many primes [10].…”

Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

“…Similarly, if E and P are defined over the function field F = K(C) of a smooth, projective, geometrically irreducible curve C over a field K, then we define the EDS of the pair (E, P ) to be the sequence of divisors D n = D nP on C defined by (1.2). See Section 1.2 for an equivalent definition in the case of perfect K. Elliptic divisibility sequences over function fields are studied in [6,8,15,28].…”